Abstract: A classic approach in dynamical systems is to use particular geometricstructures to deduce statistical properties, for example the existence ofinvariant measures with stochastic-like behaviour such as large deviations ordecay of correlations. Such geometric structures are generally highlynon-trivial and thus a natural question is the extent to which this approachcan be applied. In this paper we show that in many cases stochastic-likebehaviour itself implies that the system has certain non-trivial geometricproperties, which are therefore necessary as well as sufficient conditions forthe occurrence of the statistical properties under consideration. As a byproduct of our techniques we also obtain some new results on large deviationsfor certain classes of systems which include Viana maps and multidimensionalpiecewise expanding maps.